Factor analysis attempts to identify underlying variables, orfactors, that explain the pattern of
correlations within a set of observed variables. Factor analysis is often used in data reduction to identify a small number of factors that explain most of the variance that is observed in a much larger number of manifest variables. Factor analysis can also be used to generate hypotheses regarding causal mechanisms or to screen variables for subsequent analysis (for example, to identify collinearity prior to performing a linear regression analysis).
The factor analysis procedure offers a high degree of flexibility: v Seven methods of factor extraction are available.
v Five methods of rotation are available, including direct oblimin and promax for nonorthogonal rotations.
v Three methods of computing factor scores are available, and scores can be saved as variables for further analysis.
Example.What underlying attitudes lead people to respond to the questions on a political survey as they do? Examining the correlations among the survey items reveals that there is significant overlap among various subgroups of items--questions about taxes tend to correlate with each other, questions about military issues correlate with each other, and so on. With factor analysis, you can investigate the number of underlying factors and, in many cases, identify what the factors represent conceptually. Additionally, you can compute factor scores for each respondent, which can then be used in subsequent analyses. For example, you might build a logistic regression model to predict voting behavior based on factor scores. Statistics.For each variable: number of valid cases, mean, and standard deviation. For each factor analysis: correlation matrix of variables, including significance levels, determinant, and inverse; reproduced correlation matrix, including anti-image; initial solution (communalities, eigenvalues, and percentage of variance explained); Kaiser-Meyer-Olkin measure of sampling adequacy and Bartlett's test of sphericity; unrotated solution, including factor loadings, communalities, and eigenvalues; and rotated solution, including rotated pattern matrix and transformation matrix. For oblique rotations: rotated pattern and structure matrices; factor score coefficient matrix and factor covariance matrix. Plots: scree plot of eigenvalues and loading plot of first two or three factors.
Factor Analysis Data Considerations
Data.The variables should be quantitative at theintervalor ratiolevel. Categorical data (such as religion or country of origin) are not suitable for factor analysis. Data for which Pearson correlation coefficients can sensibly be calculated should be suitable for factor analysis.
Assumptions.The data should have a bivariate normal distribution for each pair of variables, and observations should be independent. The factor analysis model specifies that variables are determined by common factors (the factors estimated by the model) and unique factors (which do not overlap between observed variables); the computed estimates are based on the assumption that all unique factors are uncorrelated with each other and with the common factors.
To Obtain a Factor Analysis 1. From the menus choose:
Analyze>Dimension Reduction>Factor... 2. Select the variables for the factor analysis.
Factor Analysis Select Cases
To select cases for your analysis:1. Choose a selection variable.
2. ClickValueto enter an integer as the selection value.
Only cases with that value for the selection variable are used in the factor analysis.
Factor Analysis Descriptives
Statistics. Univariate descriptivesincludes the mean, standard deviation, and number of valid cases for each variable.Initial solutiondisplays initial communalities, eigenvalues, and the percentage of variance explained.
Correlation Matrix.The available options are coefficients, significance levels, determinant, KMO and Bartlett's test of sphericity, inverse, reproduced, and anti-image.
v KMO and Bartlett's Test of Sphericity. The Kaiser-Meyer-Olkin measure of sampling adequacy tests whether the partial correlations among variables are small. Bartlett's test of sphericity tests whether the correlation matrix is an identity matrix, which would indicate that the factor model is inappropriate. v Reproduced. The estimated correlation matrix from the factor solution. Residuals (difference between
estimated and observed correlations) are also displayed.
v Anti-image. The anti-image correlation matrix contains the negatives of the partial correlation
coefficients, and the anti-image covariance matrix contains the negatives of the partial covariances. In a good factor model, most of the off-diagonal elements will be small. The measure of sampling adequacy for a variable is displayed on the diagonal of the anti-image correlation matrix.
Factor Analysis Extraction
Method.Allows you to specify the method of factor extraction. Available methods are principal components, unweighted least squares, generalized least squares, maximum likelihood, principal axis factoring, alpha factoring, and image factoring.
v Principal Components Analysis. A factor extraction method used to form uncorrelated linear combinations of the observed variables. The first component has maximum variance. Successive components explain progressively smaller portions of the variance and are all uncorrelated with each other. Principal components analysis is used to obtain the initial factor solution. It can be used when a correlation matrix is singular.
v Unweighted Least-Squares Method. A factor extraction method that minimizes the sum of the squared differences between the observed and reproduced correlation matrices (ignoring the diagonals). v Generalized Least-Squares Method. A factor extraction method that minimizes the sum of the squared
differences between the observed and reproduced correlation matrices. Correlations are weighted by the inverse of their uniqueness, so that variables with high uniqueness are given less weight than those with low uniqueness.
v Maximum-Likelihood Method. A factor extraction method that produces parameter estimates that are most likely to have produced the observed correlation matrix if the sample is from a multivariate normal distribution. The correlations are weighted by the inverse of the uniqueness of the variables, and an iterative algorithm is employed.
v Principal Axis Factoring. A method of extracting factors from the original correlation matrix, with squared multiple correlation coefficients placed in the diagonal as initial estimates of the
communalities. These factor loadings are used to estimate new communalities that replace the old communality estimates in the diagonal. Iterations continue until the changes in the communalities from one iteration to the next satisfy the convergence criterion for extraction.
v Alpha. A factor extraction method that considers the variables in the analysis to be a sample from the universe of potential variables. This method maximizes the alpha reliability of the factors.
v Image Factoring. A factor extraction method developed by Guttman and based on image theory. The common part of the variable, called the partial image, is defined as its linear regression on remaining variables, rather than a function of hypothetical factors.
Analyze.Allows you to specify either a correlation matrix or a covariance matrix.
v Correlation matrix.Useful if variables in your analysis are measured on different scales.
v Covariance matrix.Useful when you want to apply your factor analysis to multiple groups with different variances for each variable.
Extract.You can either retain all factors whose eigenvalues exceed a specified value, or you can retain a specific number of factors.
Display.Allows you to request the unrotated factor solution and a scree plot of the eigenvalues. v Unrotated Factor Solution. Displays unrotated factor loadings (factor pattern matrix), communalities,
and eigenvalues for the factor solution.
v Scree plot. A plot of the variance that is associated with each factor. This plot is used to determine how many factors should be kept. Typically the plot shows a distinct break between the steep slope of the large factors and the gradual trailing of the rest (the scree).
Maximum Iterations for Convergence.Allows you to specify the maximum number of steps that the algorithm can take to estimate the solution.
Factor Analysis Rotation
Method.Allows you to select the method of factor rotation. Available methods are varimax, direct oblimin, quartimax, equamax, or promax.
v Varimax Method. An orthogonal rotation method that minimizes the number of variables that have high loadings on each factor. This method simplifies the interpretation of the factors.
v Direct Oblimin Method. A method for oblique (nonorthogonal) rotation. When delta equals 0 (the default), solutions are most oblique. As delta becomes more negative, the factors become less oblique. To override the default delta of 0, enter a number less than or equal to 0.8.
v Quartimax Method. A rotation method that minimizes the number of factors needed to explain each variable. This method simplifies the interpretation of the observed variables.
v Equamax Method. A rotation method that is a combination of the varimax method, which simplifies the factors, and the quartimax method, which simplifies the variables. The number of variables that load highly on a factor and the number of factors needed to explain a variable are minimized.
v Promax Rotation. An oblique rotation, which allows factors to be correlated. This rotation can be calculated more quickly than a direct oblimin rotation, so it is useful for large datasets.
Display.Allows you to include output on the rotated solution, as well as loading plots for the first two or three factors.
v Rotated Solution. A rotation method must be selected to obtain a rotated solution. For orthogonal rotations, the rotated pattern matrix and factor transformation matrix are displayed. For oblique rotations, the pattern, structure, and factor correlation matrices are displayed.
v Factor Loading Plot. Three-dimensional factor loading plot of the first three factors. For a two-factor solution, a two-dimensional plot is shown. The plot is not displayed if only one factor is extracted. Plots display rotated solutions if rotation is requested.
Maximum Iterations for Convergence.Allows you to specify the maximum number of steps that the algorithm can take to perform the rotation.
Factor Analysis Scores
Save as variables.Creates one new variable for each factor in the final solution.
Method.The alternative methods for calculating factor scores are regression, Bartlett, and Anderson-Rubin.
v Regression Method. A method for estimating factor score coefficients. The scores that are produced have a mean of 0 and a variance equal to the squared multiple correlation between the estimated factor scores and the true factor values. The scores may be correlated even when factors are orthogonal. v Bartlett Scores. A method of estimating factor score coefficients. The scores that are produced have a
mean of 0. The sum of squares of the unique factors over the range of variables is minimized.
v Anderson-Rubin Method. A method of estimating factor score coefficients; a modification of the Bartlett method which ensures orthogonality of the estimated factors. The scores that are produced have a mean of 0, have a standard deviation of 1, and are uncorrelated.
Display factor score coefficient matrix.Shows the coefficients by which variables are multiplied to obtain factor scores. Also shows the correlations between factor scores.
Factor Analysis Options
Missing Values.Allows you to specify how missing values are handled. The available choices are to exclude caseslistwise, exclude casespairwise, or replace with mean.
Coefficient Display Format.Allows you to control aspects of the output matrices. You sort coefficients by size and suppress coefficients with absolute values that are less than the specified value.
FACTOR Command Additional Features
The command syntax language also allows you to:v Specify convergence criteria for iteration during extraction and rotation. v Specify individual rotated-factor plots.
v Specify how many factor scores to save.
v Specify diagonal values for the principal axis factoring method.
v Write correlation matrices or factor-loading matrices to disk for later analysis. v Read and analyze correlation matrices or factor-loading matrices.